Radiative Diffusion

Our main microscopic quantity is still the specific intensity \(I_\nu\), and the main equation we solve is still the radiative transfer equation. So the diffusion picture does not replace radiative transfer. It comes out of radiative transfer in the very opaque limit.

Suppose we have a gaseous medium and an external observer looking at it. If the medium is very opaque, then what we actually observe is essentially radiation emerging from layers around optical depth of order unity. Layers much deeper than that are opaque. We cannot measure them directly. If we want to learn about them observationally, then we are forced to use indirect methods.

Now think of this \(\tau \sim 1\) layer as a kind of boundary layer that separates the opaque interior from the almost free-streaming region above it. Below that boundary, the medium may have optical depth much larger than unity. In a star, this is the usual situation. Energy is generated deep in the core by nuclear fusion, then transported all the way from those very opaque layers to the surface, and only after that does it escape to us. That is why we see the star, or in the most familiar case, why we see the Sun.

So even if we cannot directly observe the deep layers, we still need a good description of them. Otherwise we cannot build a sufficient description of the astrophysical system.

If radiation is the process that transports the energy through this opaque material, then the radiation field cannot be in complete thermodynamic equilibrium everywhere. In complete equilibrium the radiation field would be isotropic, purely blackbody, and its flux would be zero. But here the energy still has to go out. So what we need is a picture in which the radiation field is almost in equilibrium, but not exactly in equilibrium. That is the diffusion picture.

This diffusion description is the photon analogue of ordinary matter diffusion. It is also very closely related to the heat equation. Later one can connect it to a random walk of photons, or if one likes a more informal picture, to the drunkard's walk. But before getting there, we first derive the diffusion approximation directly from the radiative transfer equation.

Start again from the full radiative transfer equation written in differential form:

Now make the assumptions used in this note:

With those assumptions, the transfer equation becomes

At this stage we are not introducing the reversed optical-depth coordinate used in some plane-parallel atmosphere derivations. Here we keep the equation directly in terms of \(z\). That keeps the algebra transparent.

Now rearrange the equation so that \(I_\nu\) stands alone on one side:

Up to this point, nothing approximate has been done. The approximation enters when we say that the medium is so opaque that the radiation field differs from equilibrium only slightly. In rough language, this means optical depth much larger than unity. In more precise language, it means that the photon mean free path is small compared with the typical length scale over which temperature, density, and other microscopic variables change. We will come back to that point at the end of the chapter.

So now we say: in the diffusion picture we only have to deal with small deviations from equilibrium. The equilibrium radiation field is

This is the isotropic equilibrium situation. The source function equals the Planck function, and there is no net flux.

To obtain the first-order diffusion approximation, take the rearranged transfer equation and insert the equilibrium quantity on the right-hand side. That means: for the source term use \(B_\nu\), and in the derivative term keep only the first deviation by differentiating \(B_\nu\). Then we get

This is the first-order diffusion approximation. It says that the intensity is almost isotropic and almost equal to the Planck function, with a small correction that depends on the temperature gradient through \(B_\nu\).

We can push this further if we want. For example, one could take this expression and insert it again into the right-hand side, then obtain a second-order correction, then a third-order correction, and so on. But for the standard diffusion approximation there is no real point in doing that. The first-order form already captures the leading departure from equilibrium, and that is the form we now use.

Before doing the angular integrals, keep one technical point in mind. The remaining results below assume that \(\alpha_\nu\) is angle independent, or isotropic. In a static medium this is fine. If the medium had a bulk velocity, then in general the opacity would acquire an angle dependence. In that case one usually transforms to the frame co-moving with the fluid and does the calculation there.

3.1 Energy Density

The monochromatic radiation energy density is

Now insert the diffusion approximation, and use \(d\Omega = 2\pi\,d\mu\) because there is no azimuthal dependence:

The second term vanishes, because it contains one power of \(\mu\) integrated symmetrically from \(-1\) to \(1\). That is an odd integrand. So only the first term survives:

Now integrate over frequency:

So the energy density is unchanged in the diffusion approximation. The same is true for the mean intensity, since it is directly proportional to the energy density.

3.2 Radiation Pressure

Now take the second moment. In this one-dimensional Cartesian setting, the relevant pressure component is the \(zz\) component:

Again, the correction term vanishes. This time it contains \(\mu^3\), which is also odd and therefore integrates to zero over the symmetric interval. So only the equilibrium part remains:

So the second moment is also unchanged. The diffusion approximation does not alter the equilibrium relation between radiation energy density and radiation pressure.

3.3 Flux

Now take the first moment. This is the one we were waiting for, because this is the moment that gives the energy flux. In complete equilibrium the flux is zero. Here it will not vanish, because the small first-order departure from isotropy is exactly what carries the energy.

The first term now vanishes, because it contains only one power of \(\mu\) multiplying an angle-independent \(B_\nu\). But the correction term survives, because it gives \(\mu^2\):

Now apply the chain rule:

This is the key result. The flux is proportional to minus the temperature gradient. In other words, radiative diffusion has exactly the form of a heat-transport law.

The minus sign matters. If the flux points outward and we take \(z\) to increase outward, then \(F_{\nu,z} > 0\) means \(dT/dz < 0\). So the temperature must decrease outward, or said in the simpler way used in the lecture: the temperature increases inward. That is the simplest way to understand why deeper layers are hotter if radiation is transporting energy outward through an opaque medium.

Integrate the monochromatic flux over frequency:

This integral is not trivial, because the opacity can depend strongly on frequency. But one physical point already stands out very clearly: the contribution is weighted by \(1/\alpha_\nu\). That means lower-opacity frequency ranges carry more of the diffusive flux, while very opaque frequency ranges carry less.

A simple picture helps. Imagine a muddy pond. If the mud is spread out everywhere, then light from below hardly escapes. Now take the same total amount of mud and push most of it into one corner. Suddenly there are clearer parts of the pond through which light can slip out. The total amount of obscuring material has not changed, but its distribution matters. In radiative diffusion, the same thing happens in frequency space: low-opacity windows let more radiation escape, while very opaque bands block it. One can say that the medium becomes porous in frequency space.

If the total radiative flux still has to come out, then blocking in one frequency range leads to redistribution into other frequency ranges. In stellar atmospheres one often speaks of line blocking in the ultraviolet and redistribution toward frequency ranges where the opacity is lower. The same basic diffusion idea is at work here.

Now define the opacity in the usual way:

Here \(\alpha_\nu\) has units of inverse length, \(\rho\) is the mass density, and \(\kappa_\nu\) is the mass absorption coefficient. The units are therefore

To write the flux compactly, introduce the Rosseland mean opacity. First note that the denominator we need is simply

Now define the weighted harmonic mean:

Equivalently, because \(\rho\) is not frequency dependent, one may write

With this definition, the frequency-integrated flux becomes

Now use the integrated Planck function \(B=\sigma T^4/\pi\). Then

Substitute this back in:

This is often written in the compact conduction-like form

where \(K_R\) is the radiative conductivity.

We can also rewrite the flux in terms of the radiation energy density. Since

we have

This is already the standard diffusion form: flux is proportional to minus the gradient of the transported quantity.

Define the mean free path corresponding to the Rosseland mean opacity by

Frequency by frequency, the corresponding monochromatic mean free path is

With \(\ell\) introduced, the diffusion flux can be written as

This makes the meaning very clear. The radiative flux is driven by the gradient of the radiation energy density, and the coefficient multiplying that gradient is set by the speed of light together with the photon mean free path. This is why radiative diffusion really is a diffusion process.

So the practical condition for the diffusion approximation is not just “optical depth larger than one” in a vague sense. The sharper condition is that the photon mean free path must be small compared with the scale over which the physical state of the medium changes. Only then does the radiation field remain very close to local equilibrium, with only small first-order departures.

This is exactly the regime in which the diffusion picture becomes useful for astrophysical interiors: very opaque media, small mean free paths, nearly isotropic radiation, but still a small anisotropy large enough to carry a net outward flux.

And that is where this part naturally stops. The next step is to connect this mean free path to a random walk of photons and then to the heat-equation form of diffusion. That gives another way of understanding the same result, but the basic diffusion approximation itself is already complete here.